# Gentle(-er) Introduction to Erdős–Bollobás's solution to Ramsey–Turán Type Problem

I am currently trying to understand the construction of maximal graph which contains no $$K_4$$ and sub-linear number of independent points in the graph. The original paper On a Ramsey–Turán type problem, although ground-breaking, is very hard to parse. It also glosses over some crucial details and claims some really non-trivial identities. I am listing some of them below (but there are probably more questions one can ask).

• Is there a way to partition a hyper-sphere of dimension $$n$$ into $$N$$ equal parts and bounded diameters?
• Aren't $$C$$ and $$A$$, as defined in the paper, very close to each other? Importantly, doesn't that imply that $$\delta > 1$$ (and that's it)?
• Can anyone explain the argument made to find the diameter of the spherical cap that, as claimed on top of page 3?
• Note: Erdős's name is spelled with ő ("o with double acute"), not ö ("o with diaeresis"). I have edited accordingly. I also edited out an institutionally gated link to the article, and replaced it with a DOI link. Oct 15 at 19:18
• @LSpice Thank you, I forgot about the gated link :) Oct 16 at 4:08
• Re, no worries. Is there a missing word or words in your third bullet point ("… the diameter of the spherical cap that, as claimed …")? Oct 16 at 11:31
• An alternative presentation of the Bollobás-Erdős construction appears in this paper by Balogh and Lenz (sections 3, 4) arxiv.org/pdf/1109.4428.pdf
– hdur
Oct 18 at 13:32
• Name of the paper referenced by @hdur: Balogh and Lenz - On the Ramsey–Turán numbers of graphs and hypergraphs. Oct 18 at 19:22

## 1 Answer

I found an excellent account of this problem in the thesis of John Lenz: xtremal graph theory: Ramsey-Turán numbers, chromatic thresholds, and minors Lenz, John E.